The Relation between the Modified Korteweg-de Vries Equation and an Anomaly of the Dirac Field on a Thin Elastic Rod

Abstract

Recently a study on the Dirac equation in a thin elastic rod as a submanifold quantum system was reported. (S. Matsutani and H. Tsuru, Phys. Rev. A46 (1992), 1144). It is shown that the Dirac operator is identified with the Lax operator of the Modified Korteweg-de Vries (MKdV) equation. Since the thin elastic rod is governed by the MKdV equation, it implies that the fictitious quantum mechanics in the soliton physics has a real physical meaning as a fermionic field on the elastic rod. In this paper, we quantize the Dirac field in a thin elastic rod on R • We find that there appears an anomalous current proportional to the curvature of the rod which is the MKdV soliton. Hence we prove that the relation between the fermionic field systell). and the classical MKdV soliton is regarded as an anomaly on the thin elastic rod.